Chapter 8 Statistics

8.1 Descriptive statistics

8.1.1 Definitions

• • Population : set of similar items

• • Sample : subset of the population

• • Sample size \(n\) : total number of measurements

• • Variable \(x\) : measure

8.1.2 Variables

• • Continuous: gaussian distribution

• • Discrete: ordinal, nominal or binomial

8.1.3 Measures

Central tendency

• • mean \(\bar {x}\)

  \[\bar {x} = \dfrac {1}{n} \sum _i x_i\]

• • median \(Med\) : middle

• • mode \(Mod\) : most occurring

Dispersion

• • range \(Ran\) : difference between the largest and smallest values

  \[Ran = x_{\text {max}} - x_{\text {min}}\]

• • variance \(Var\) :

  \[Var_x = \dfrac {1}{n} \sum _i \left (x_i - \bar {x}\right )^2\]

• • standard deviation \(s\) :

  \[s_x = \sqrt {\dfrac {1}{n} \sum _i \left (x_i - \bar {x}\right )^2}\]

8.2 Combinatorics

8.2.1 Product rule

\(r\) independent partial decisions with

\[n_1, n_2, \ldots , n_r\]

possibilities

\[\Rightarrow \#\text {possibilities} = n_1 \cdot n_2 \cdot \ldots \cdot n_r\]

8.2.2 Sum rule

With double counting

\[\#(A \cup B) = \#A + \#B - \#(A \cap B)\]

• • \(\#A\) : number of elements in \(A\)

• • \(A \cup B\) : union, \(A\) or \(B\)

• • \(A \cap B\) : intersection, \(A\) and \(B\)

Mutually exclusive possibilities If \(A\) and \(B\) are disjoint sets

\[ \Rightarrow \#(A\cup B) = \#A + \#B\]

8.2.3 Factorials

Definition

\begin{align*} \forall n \in \mathbb {N}\backslash \{0,1\}: n! &= n\cdot (n-1)\cdot (n-2)\cdot \cdots \cdot 3\cdot 2\cdot 1\\ 1! &= 1\\ 0! &= 1\\ \end{align*}

Properties

\begin{align*} (n+1)! &= (n+1) \cdot n!\\ n! &= n \cdot (n-1)!\\ \end{align*}

8.2.4 Permutations

\begin{align*} P_n &= n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1\\ &= n! \end{align*}

8.2.5 K-Permutations

\begin{align*} P^k_n &= n \cdot (n-1) \cdot (n-2) \cdots (n-k+1)\\ &= \dfrac {n!}{(n-k)!} \end{align*}

8.2.6 K-Permutations with repetitions

\[\bar {V}^k_n=n^k\]

8.2.7 Combinations

\[C^k_n=\dfrac {n!}{k!\cdot (n-k)!}\]

8.2.8 Combinations with repetitions

\[\bar {C}^k_n=C^k_{n+k-1}\]

8.2.9 Anagrams

\begin{align*} P^{n_1,n_2,n_3,\dots ,n_k}_n &= \dfrac {n!}{n_1!\cdot n_2!\cdot n_3!\cdot \cdots \cdot n_k!}\\ &=\dfrac {\left (\sum _{i=1}^k n_i\right )!}{\prod _{i=1}^k n_i!} \end{align*}

8.2.10 Pascal’s triangle

Binomialcoefficients

the binomiaal coefficient where from \(n\) elements we choose \(k\):

\[{n \choose k} = \dfrac {n!}{k! \cdot (n-k)!}\]

Recursive formula

If \(k,n \in \mathbb {N}\) and \(k \lt n\):

\[{n \choose k} = {n-1 \choose k} + {n-1 \choose k-1}\]

Triangle

\[ \scriptsize \begin {array}{r|cccccccccccccc} \underset {\downarrow }{n}\\ 0 & 1 \\ 1 & 1 & 1 \\ 2 & 1 & 2 & 1 \\ 3 & 1 & 3 & 3 & 1 \\ 4 & 1 & 4 & 6 & 4 & 1 \\ 5 & 1 & 5 & 10 & 10 & 5 & 1 \\ 6 & 1 & 6 & 15 & 20 & 15 & 6 & 1 \\ 7 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 \\ 8 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 \\ 9 & 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 \\ 10 & 1 & 10 & 45 & 120 & 210 & 252 & 210 & 120 & 45 & 10 & 1 \\ 11 & 1 & 11 & 55 & 165 & 330 & 462 & 462 & 330 & 165 & 55 & 11 & 1 \\ 12 & 1 & 12 & 66 & 220 & 495 & 792 & 924 & 792 & 495 & 220 & 66 & 12 & 1 \\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & \leftarrow k\\ \end {array} \]

Properties

\[{n \choose k} = {n \choose n-k}\]

Binomium of Newton

\[(a+b)^n=\sum ^n_{k=0}{n \choose k}a^{n-k}b^k\]